| Library: | kalman |
| See also: | gkalarray gkallag gkalsmoother gkalresiduals |
| Quantlet: | gkalfilter | |
| Description: | Calculates a filtered time series for a state space model (uni- or multivariate) with time variable system matrices using the Kalman filter. Furthermore, gkalfilter gives the value of the log likelihood function. |
| Usage: | {gkalfilOut,loglike} = gkalfilter(Y,mu,Sig,ca,Ta,Ra,da,Za,Ha,l) | |
| Input: | ||
| Y | N_max x TIME matrix of observed time series, N_max is the maximal number of variables observed at any instant t | |
| mu | K x 1 vector, the mean of the initial state | |
| Sig | K x K covariance matrix of the initial state | |
| ca | K x 1 x TIME array with observations c_t | |
| Ta | K x K x TIME array with observations T_t | |
| Ra | K x K x TIME array with covariance matrices R_t | |
| da | N_max x 1 x TIMES array with observations d_t | |
| Za | N_max x K x TIMES array with observations Z_t | |
| Ha | N_max x N_max x TIMES array with covariance matrices H_t | |
| l | set to 1 if the value of the log likelihood function should be displayed. Otherwise, you can omit this variable. | |
| Output: | ||
| gkalfilOut | K x (K+1) x (TIME+1) array of state space series a_t~P_t. First entry is mu~Sig for t=0. | |
| loglike | value of the log likelihood function evaluated with the results of the filter. | |
State equation
alpha_t = c_t + T_t alpha_t-1 + e^s_t
Measurement equation
y_t = d_t + Z_t alpha_t + e^m_t
with
alpha_0 ~ (mu,Sig), e^s_t ~ (0,R_t), e^m_t ~ (0,H_t)
All parameters are known.
We denote with a_t the filtered series of the unobservable state vector alpha_t and with P_t the corresponding covariance matrix. The output of the procedure is an array with a_t and P_t for all t.
gkalfilter treats missing values in the following way:
1. missing values in the measurement matrix Y will be removed by gkalfilter.
2. there must be no missing values in the matrices belonging to the state equation if the corresponding observations in Y are non-missing.
If you are not interested in the value of the log likelihood function, you should omit l. This makes the procedure faster.
library("kalman")
library("xplore") ; loads the quantlets from xplore library
library("plot") ; loads the quantlets from plot library
Y = read("houseprice.dat")
mu = #(33,-13,1,1,1)
Sig = 0.75*unit(5)
ca = gkalarray(Y,#(0.1,0,0,0,0),0,0)
T =(#(1.4,1)'|#(-0.4,0)')~0*matrix(2,3)|(0*matrix(3,2)~unit(3))
Ta = gkalarray(Y,T,0,0)
Ra = gkalarray(Y,diag(#(0.4,0,0,0,0)),0,0)
da = gkalarray(Y,#(0,0,0),0,0)
Z = matrix(3)~0*matrix(3,4)
IZ = 0*matrix(3,2)~(#(1:3)'|#(4:6)'|#(7:9)')
XZ = read("housequality.dat")
Za = gkalarray(Y,Z,IZ,XZ)
Ha = gkalarray(Y,2*unit(3),0,0)
{gkalfilOut,loglike} = gkalfilter(Y,mu,Sig,ca,Ta,Ra,da,Za,Ha)
Time =(1:cols(Y))
T = rows(Time)
state = reshape(gkalfilOut[1,1,2:(T+1)],#(1,T,1))'
state = Time~state
state = setmask(state,"line","blue","medium")
Y1 = Time~(Y[1,]')
setmaskp(Y1,0,8,5)
Y2 = Time~(Y[2,]')
setmaskp(Y2,0,8,5)
Y3 = Time~(Y[3,]')
setmaskp(Y3,0,8,5)
disp = createdisplay(1,1)
show(disp,1,1,state,Y1,Y2,Y3)
setgopt(disp,1,1,"title","Kalman filter","xlabel","Time","ylabel","y,a","border",0)
setgopt(disp,1,1,"ylabel","y,a","border",0)
The filtered common price component (i.e., the first component of the filtered state vector) is shown as a blue line; the observations of Y are shown as black dots.